3.190 \(\int \frac{1}{\sqrt{a+b \cos ^{-1}(c x)}} \, dx\)

Optimal. Leaf size=102 \[ \frac{\sqrt{2 \pi } \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{\sqrt{b} c}-\frac{\sqrt{2 \pi } \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{\sqrt{b} c} \]

[Out]

-((Sqrt[2*Pi]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]])/(Sqrt[b]*c)) + (Sqrt[2*Pi]*Fres
nelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[a/b])/(Sqrt[b]*c)

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Rubi [A]  time = 0.0901623, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4624, 3306, 3305, 3351, 3304, 3352} \[ \frac{\sqrt{2 \pi } \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{\sqrt{b} c}-\frac{\sqrt{2 \pi } \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{\sqrt{b} c} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a + b*ArcCos[c*x]],x]

[Out]

-((Sqrt[2*Pi]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]])/(Sqrt[b]*c)) + (Sqrt[2*Pi]*Fres
nelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[a/b])/(Sqrt[b]*c)

Rule 4624

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[1/(b*c), Subst[Int[x^n*Sin[a/b - x/b], x], x, a
+ b*ArcCos[c*x]], x] /; FreeQ[{a, b, c, n}, x]

Rule 3306

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d +
f*x]/Sqrt[c + d*x], x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/Sqrt[c + d*x], x], x] /; FreeQ[{c
, d, e, f}, x] && ComplexFreeQ[f] && NeQ[d*e - c*f, 0]

Rule 3305

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Sin[(f*x^2)/d], x], x,
Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3351

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rule 3304

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Cos[(f*x^2)/d],
x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3352

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{a+b \cos ^{-1}(c x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sin \left (\frac{a}{b}-\frac{x}{b}\right )}{\sqrt{x}} \, dx,x,a+b \cos ^{-1}(c x)\right )}{b c}\\ &=-\frac{\cos \left (\frac{a}{b}\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{x}{b}\right )}{\sqrt{x}} \, dx,x,a+b \cos ^{-1}(c x)\right )}{b c}+\frac{\sin \left (\frac{a}{b}\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{x}{b}\right )}{\sqrt{x}} \, dx,x,a+b \cos ^{-1}(c x)\right )}{b c}\\ &=-\frac{\left (2 \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \cos ^{-1}(c x)}\right )}{b c}+\frac{\left (2 \sin \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \cos ^{-1}(c x)}\right )}{b c}\\ &=-\frac{\sqrt{2 \pi } \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{\sqrt{b} c}+\frac{\sqrt{2 \pi } C\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right ) \sin \left (\frac{a}{b}\right )}{\sqrt{b} c}\\ \end{align*}

Mathematica [C]  time = 0.081981, size = 118, normalized size = 1.16 \[ \frac{e^{-\frac{i a}{b}} \left (\sqrt{-\frac{i \left (a+b \cos ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{i \left (a+b \cos ^{-1}(c x)\right )}{b}\right )+e^{\frac{2 i a}{b}} \sqrt{\frac{i \left (a+b \cos ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},\frac{i \left (a+b \cos ^{-1}(c x)\right )}{b}\right )\right )}{2 c \sqrt{a+b \cos ^{-1}(c x)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/Sqrt[a + b*ArcCos[c*x]],x]

[Out]

(Sqrt[((-I)*(a + b*ArcCos[c*x]))/b]*Gamma[1/2, ((-I)*(a + b*ArcCos[c*x]))/b] + E^(((2*I)*a)/b)*Sqrt[(I*(a + b*
ArcCos[c*x]))/b]*Gamma[1/2, (I*(a + b*ArcCos[c*x]))/b])/(2*c*E^((I*a)/b)*Sqrt[a + b*ArcCos[c*x]])

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Maple [A]  time = 0.056, size = 85, normalized size = 0.8 \begin{align*} -{\frac{\sqrt{2}\sqrt{\pi }}{c}\sqrt{{b}^{-1}} \left ( \cos \left ({\frac{a}{b}} \right ){\it FresnelS} \left ({\frac{\sqrt{2}}{\sqrt{\pi }b}\sqrt{a+b\arccos \left ( cx \right ) }{\frac{1}{\sqrt{{b}^{-1}}}}} \right ) -\sin \left ({\frac{a}{b}} \right ){\it FresnelC} \left ({\frac{\sqrt{2}}{\sqrt{\pi }b}\sqrt{a+b\arccos \left ( cx \right ) }{\frac{1}{\sqrt{{b}^{-1}}}}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*arccos(c*x))^(1/2),x)

[Out]

-2^(1/2)*Pi^(1/2)*(1/b)^(1/2)*(cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)/(1/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)-sin(a
/b)*FresnelC(2^(1/2)/Pi^(1/2)/(1/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b))/c

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b \arccos \left (c x\right ) + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*arccos(c*x))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(b*arccos(c*x) + a), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*arccos(c*x))^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a + b \operatorname{acos}{\left (c x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*acos(c*x))**(1/2),x)

[Out]

Integral(1/sqrt(a + b*acos(c*x)), x)

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Giac [A]  time = 1.69054, size = 220, normalized size = 2.16 \begin{align*} \frac{\sqrt{\pi } i \operatorname{erf}\left (-\frac{\sqrt{2} \sqrt{b \arccos \left (c x\right ) + a} i}{2 \, \sqrt{{\left | b \right |}}} - \frac{\sqrt{2} \sqrt{b \arccos \left (c x\right ) + a} \sqrt{{\left | b \right |}}}{2 \, b}\right ) e^{\left (\frac{a i}{b}\right )}}{{\left (\frac{\sqrt{2} b i}{\sqrt{{\left | b \right |}}} + \sqrt{2} \sqrt{{\left | b \right |}}\right )} c} + \frac{\sqrt{\pi } i \operatorname{erf}\left (\frac{\sqrt{2} \sqrt{b \arccos \left (c x\right ) + a} i}{2 \, \sqrt{{\left | b \right |}}} - \frac{\sqrt{2} \sqrt{b \arccos \left (c x\right ) + a} \sqrt{{\left | b \right |}}}{2 \, b}\right ) e^{\left (-\frac{a i}{b}\right )}}{{\left (\frac{\sqrt{2} b i}{\sqrt{{\left | b \right |}}} - \sqrt{2} \sqrt{{\left | b \right |}}\right )} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*arccos(c*x))^(1/2),x, algorithm="giac")

[Out]

sqrt(pi)*i*erf(-1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sqrt(
abs(b))/b)*e^(a*i/b)/((sqrt(2)*b*i/sqrt(abs(b)) + sqrt(2)*sqrt(abs(b)))*c) + sqrt(pi)*i*erf(1/2*sqrt(2)*sqrt(b
*arccos(c*x) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sqrt(abs(b))/b)*e^(-a*i/b)/((sqrt(2)*b*
i/sqrt(abs(b)) - sqrt(2)*sqrt(abs(b)))*c)